As only the outer surface of the sphere is charged, consider...

As only the outer surface of the sphere is charged, consider the element as a ring, as shown in the figure. The equivalent current due to the ring element, $d i=\frac{\omega}{2 \pi}(2 \pi r \sin \theta n d \theta) \sigma$ and magnetic induction due to this loop element at the centre of the sphere, $O$, $d B=\frac{\mu_{0}}{4 \pi} d i \frac{2 \pi r \sin \theta r \sin \theta}{r^{3}}=\frac{\mu_{0}}{4 \pi} d i \frac{\sin ^{2} \theta}{r}$ [Using $3.219$ (b) ] Hence, the total magnetic induction due to the sphere at the centre, $O$, $\pi / 2$ $B=\int d B=\int_{0} \frac{\mu_{0}}{4 \pi} \frac{\omega}{2 \pi} \frac{2 \pi r^{2} \sin \theta d \theta \sin ^{2} \theta \sigma}{r}$ [using (1)] $\pi / 2$ Hence, $\quad B=\int_{0} \frac{\mu_{0} \sigma \omega r}{4 \pi} \sin ^{3} \theta d \theta=\frac{2}{3} \mu_{0} \sigma \omega r=29 \mathrm{pT}$

As only the outer surface of the sphere is charged, consider the element as a ring, as shown in the figure. The equivalent current due to the ring element,
$d i=\frac{\omega}{2 \pi}(2 \pi r \sin \theta n d \theta) \sigma$
and magnetic induction due to this loop element at the centre of the sphere, $O$, $d B=\frac{\mu_{0}}{4 \pi} d i \frac{2 \pi r \sin \theta r \sin \theta}{r^{3}}=\frac{\mu_{0}}{4 \pi} d i \frac{\sin ^{2} \theta}{r}$
[Using $3.219$ (b) ] Hence, the total magnetic induction due to the sphere at the centre, $O$, $\pi / 2$
$B=\int d B=\int_{0} \frac{\mu_{0}}{4 \pi} \frac{\omega}{2 \pi} \frac{2 \pi r^{2} \sin \theta d \theta \sin ^{2} \theta \sigma}{r}$ [using (1)]
$\pi / 2$
Hence, $\quad B=\int_{0} \frac{\mu_{0} \sigma \omega r}{4 \pi} \sin ^{3} \theta d \theta=\frac{2}{3} \mu_{0} \sigma \omega r=29 \mathrm{pT}$

Key Concepts

Rotating Charged Sphere

When a sphere carrying a charge only on its surface rotates, the motion of the charges produces currents that generate a magnetic field. This concept bridges the study of static charge distributions and dynamic systems where charge movement results in magnetism, illustrating a fundamental aspect of electromagnetism.

Equivalent Current Distribution

A rotating charged surface can be analyzed by considering its charge elements as forming current loops. By decomposing the charged sphere into differential ring elements, each with its specific contribution to the magnetic field, the overall magnetic effect is obtained through integration. This method transforms a complex distributed current into a sum of simpler current elements.

Biot–Savart Law

This fundamental principle relates the magnetic field generated by an element of current to the magnitude, direction, and position of that current element. In problems involving continuous current distributions, such as the rotating charged sphere, the Biot–Savart Law is applied to each differential element, and the contributions are integrated to yield the net magnetic field.

Surface Charge Density

Surface charge density defines the amount of electric charge per unit area on the surface of a conductor. For a rotating sphere, the surface charge density directly influences the strength of the equivalent current generated due to the motion of the charges, thereby playing a critical role in determining the resultant magnetic field.

Differential Integration in Spherical Coordinates

Calculating the total magnetic effect from a rotating charged sphere involves breaking down the surface into infinitesimal elements, often expressed in terms of spherical coordinates. By integrating the contributions from these elements—organized by the polar angle—it is possible to account for the varying geometry and ensure that all parts of the spherical surface are included in the final magnetic field calculation.

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